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Metamath Proof Explorer

Theorem opoc0

Description: Orthocomplement of orthoposet zero. (Contributed by NM, 24-Jan-2012)

Ref Expression
Hypotheses opoc1.z 
|- .0. = ( 0. ` K )
opoc1.u 
|- .1. = ( 1. ` K )
opoc1.o 
|- ._|_ = ( oc ` K )
Assertion opoc0 
|- ( K e. OP -> ( ._|_ ` .0. ) = .1. )

Proof

Step Hyp Ref Expression
1 opoc1.z 
 |-  .0. = ( 0. ` K )
2 opoc1.u 
 |-  .1. = ( 1. ` K )
3 opoc1.o 
 |-  ._|_ = ( oc ` K )
4 1 2 3 opoc1 
 |-  ( K e. OP -> ( ._|_ ` .1. ) = .0. )
5 eqid 
 |-  ( Base ` K ) = ( Base ` K )
6 5 2 op1cl 
 |-  ( K e. OP -> .1. e. ( Base ` K ) )
7 5 1 op0cl 
 |-  ( K e. OP -> .0. e. ( Base ` K ) )
8 5 3 opcon1b 
 |-  ( ( K e. OP /\ .1. e. ( Base ` K ) /\ .0. e. ( Base ` K ) ) -> ( ( ._|_ ` .1. ) = .0. <-> ( ._|_ ` .0. ) = .1. ) )
9 6 7 8 mpd3an23 
 |-  ( K e. OP -> ( ( ._|_ ` .1. ) = .0. <-> ( ._|_ ` .0. ) = .1. ) )
10 4 9 mpbid 
 |-  ( K e. OP -> ( ._|_ ` .0. ) = .1. )